
with \(r=\lfloor 2(n-|d|)/3\rfloor .\)  Liu and Petrov [1]} proved that, for any positive integer \(n\) ,
\(\sum _{k=0}^{n-1}q^k {\genfrac[]{0.0pt}{}{2k}{k}}_{}\equiv \left(\frac{n}{3}\right)q^{(n^2-1)/3}\ ({\rm {mod}}\ \Phi _n(q)^2),\) 
